1. Contiguous area cartograms Ingeborg Groeneweg

2. Introduction What are cartograms Difficulties creating cartograms History: previous approaches Current approach in-depth Summary

3. Cartograms Resizing regions of map by geographically related parameter Other way of representing the same: –3D-map –choropleth

4. Cartograms Area resizing:  x  y = h  u  v Shape preservation Topology preservation Contiguous

5. Example

6. 2 parameter example

7. Difficulties Resizing area, preserving shape, preserving angle

8. Error measure Optimization problem Error measure –Area error –Shape error

9. History: Rubber map method Tobler, 1973 One of the first cartogram algorithms Used for population districting Idea: –put a dot for every person on a rubber map –Stretch the map until every dot is at equal distance Problem: –poor performance –Large area error –Overlapping shapes

10. Pseudo cartogram Tobler, 1986 Reduce area error Starting point for rubber map method

11. Rubber sheet distortion Dougenik, Chrisman, Niemeyer, 1985 Improvement on Tobler Difference: computing “force” on one polygon per iteration Overlapping shapes occur infrequently

12. DEMP Selvin et al.,1984 Density Equalized Map Projection (DEMP) Used to detect non-random distributions of disease Calculate spatial magnificent factor Radial transformation projected on selected area

13. Line integral Gusein-zade and Tikunov, 1995 Stokes theorem and line integrals

14. Forced-based Kocmoud and House, 1998 Alternately optimize shape and area error Superior to former methods

15. Cartodraw Keim, north, panse, 2004 Goal: –creating cartograms on the fly –Small error Cartodraw: –Simplify shape –Define error functions –Scanline –Main algorithm

16. Cartodraw: simplify shape Few vertices important for defining shape Vertices almost no noticeable difference: –angle near 180 degree –With short edges Different reduction algorithm for global shape and inner vertices

17. Global polygon reduction Only look at vertices v with d( v )> f( v ) –v at outer polygon –v do not belong tomultiple polygons Determine least important vertex w Finding polygon p where w is part of Counting difference d between p before and after removing w Remove w if d < constant

18. Least important vertex

19. Inner polygon reduction Remove all interior vertices v with d( v ) = 2 Reintroduce few vertices

20. Cartodraw Area error function Shape similarity function Scanline algorithm Main algorithm

21. Area error function Relative area error of polygon p j Area error for set of polygons P

22. Shape error function Translation, scale and partially rotation invariant Euclidean distance in Fourier space useful for shape similarity measure Use of differential geometric curvature of polygons Curvature will be square wave function

23. curvature

24. Example curvature

25. Fourier transformation Approximate function by summing sine and cosine Fourier approximation is defined as:

26. Fourier of square wave

27. Scanline Scanline sl = Line segment of arbitrary position and length Incrementally reposition vertices along scanline

28. Scanline Scaling factor

29. Cartodraw: main algorithm Transformation applied for each scanline –If E rel and shape distortion below certain threshold changes are retained Test improvement of area error

30. Automatic vs interactive Automatic generation of scanlines: –Fixed grid of horizontal and vertical scanlines –Resolution can be varied Interactive position of scanlines –Better results

31. Summary No ideal solution Several approaches reviewed Poor performance Handmade solutions superior